Systemc Behavour of Plane Recprocal Frame Structures Thomas Kohlhammer, Char of Structural Desgn, ETH Zurch, Zurch, Swtzerland; Ton Kotnk, Char of Structural Desgn, ETH Zurch, Zurch, Swtzerland Contact: kohlhammer@arch.ethz.ch DOI:.2749/6866X292572596 bstract Recprocal frame constructon refers to a structural system that was developed frst n eastern sa n the 2th century. Short bar-shaped elements allow a surface to be spanned whose area s many tmes that of the length of the ndvdual bars. In addton to the global geometry of the resultng surface, of partcular nterest s the nteracton of forces between the ndvdual bars that enables the load support and gves rse to the specfc systemcs of the overall structure. Ths paper ntends to analyse these topcs and the resultng possbltes. Keywords: Dscrete structures; bar-shaped structures; spatal structures; structural analyss; analyss method. Introducton recprocal frame s understood as a structural system formed by a number of short bars that are connected usng frcton only and span many tmes the length of the ndvdual bars (Fg. a). Ths paper descrbes an academc vew of the structural behavour of such systems. method s presented that descrbes the dstrbuton of forces through the structure and can serve as a bass of a desgn method for practcal usage. Hstory The frst recprocal frame structures appeared n Chnese and Japanese archtecture n the 2th century, prmarly wth wood-constructed roof support system descrbed as the mandala roof (Fg. b). Today, ths roof desgn and also more complex recprocal frame desgns (Fg. d) are manly used n Japanese archtecture (see, e.g., Kazuhro Ish 2 and Shgeru Ban ). In Europe, recprocal frame structures were frst ntroduced n the th century by the gothc archtect Vllard de Honnecourt. 4 Hs sketch books show llustratons of suggestons for roof support systems usng ths desgn prncple. In the begnnng of the 6th century, Leonardo da Vnc developed dea sketches for brdges, roofs and celngs accordng to the prncple of recprocal frames, 5 and Sebastano Serlo addressed the problem of how to span a celng wth beams that were sgnfcantly shorter than the span of the celng tself (Fg. 2a). 6 comparable structure system made of recprocally supportng bar-shaped elements s the Zollnger System whch s manly used n tmber roof constructons. 7 Fredrch Zollnger obtaned a patent for t n 92. (c) Sophstcated tmber products such as glulam trusses and plywood that produce long spannng structural elements through adhesve technology have led to the replacement of recprocal frames and smlar structures. However, these hghly nterestng supportng structures can be found today n academc envronment where the system contnues to be used as an expermental model due to the fascnatng smplcty of the recprocally bearng elements. In addton, research nterest s motvated by the fact that comparable structures of dscrete elements do occur n nature, for example n brd nests. In the course of the ever-ncreasng sgnfcance of bomorphc archtectural language, t s mmedately mportant to become acquanted wth the functonal behavour of such structures. 8 In the course of awareness of recyclablty and resource savng, recprocal (d) Peer-revewed by nternatonal experts and accepted for publcaton by SEI Edtoral Board Paper receved: March 5, 2 Paper accepted: ugust 9, 2 Fg. : Smple recprocal frame structure; Bunraku Puppet Theatre Exhbton Hall, Sewa, Kazuhro Ish, 994 2 (S. 9); (c) more complex recprocal frame structure 2 (S. 2); (d) Bunraku Puppet Theatre udtorum, Sewa, Kazuhro Ish, 994 2 (S. ) 8 Scentfc Paper Structural Engneerng Internatonal /2
(c) (d) < produces a concave curvature, and a gradent = of each bar results n a plane system (Fg. 2b). In a horzontal plane recprocal frame structure, the ndvdual bars can be assumed to be statcally determnate sub-systems, due to the fact that they act as a smple beam. The supportng forces of each bar are thus ndependent of the materal and bar cross secton. Wth regard to horzontal forces, a three-bar basc element s statcally determnate nternally to the extent that the ntersectons are assumed to be flexble connectons. Hgher barred elements or elements consstng of more than three bars are moveable. The statc determnacy of the supportng structure n regard to horzontal load can be controlled by approprate combnatons of hgher barred basc elements and by approprately supportng the entre structure. Fg. 2: Sebastano Serlos proposal of a celng structure composed of short elements 6 (S. ); plane recprocal frame structure; (c) examples of basc elements: Regular (left), rregular (rght); (d) examples of recprocal frame structures consstng of dentcal basc elements (left), consstng of non-dentcal basc elements (rght) frames and smlar structures should be consdered agan. If the nternal forces are known, each element n such structures can be adjusted to ts stress and therefore an optmzed materal consumpton can be guaranteed. Nowadays, such customzaton s easly realzable through the possblty of dgtal fabrcaton. Geometry and Load-Bearng Behavour recprocal frame structure s decomposable nto basc elements whch crcumscrbe a polygon wth at least three sdes whereby the fgure may be ether regular or rregular (Fg. 2c). recprocal frame structure can be constructed from dentcal or non-dentcal basc elements as long as a tessellaton pattern exsts (Fg. 2d). The jonng of the elements at the node ponts can generally be carred out wthout mechancal connectons, but solely by pressure and frcton. To support the frctonal force, smple connecton technques such as tyng together (Fg. c) or notchng of the bars at the contact ponts may be used (Fg. 2b). Drectly dependent on the development of the connectons s the deformaton of the entre system under loadng. Increased slppage that occurs wth smple jonng, such as tyng together, results n ncreased deformablty of the entre structure. From a structural standpont, each ndvdual bar n the system functons as a sngle beam. Ths beam les at each of the bar s ends ether on another bar or, f t forms the edge of the system, on the supports of the entre system. Each bar bears the supportng force of the one or two bars restng on t and optonal dead loads or lve loads (Fg. 7). Wth straght bars lyng on top of each other at ther node ponts, the heght offset between the contact ponts of the actng forces and the supportng loads creates a convex curvature of the system (Fg. c). In ths case, the degree of curvature can be controlled by the gradents of bar length and the dstance between the actng force and the supportng force. ddtonal optons for controllng the gradents exst n the form of bendng of the bars or notchng at the node ponts, whereby gradent values can be acheved. gradent Systemc Observaton In the followng text, the nteracton of the elements of recprocal frame structures under loadng wll be addressed. Only plane structures under perpendcular loadng wll be consdered. The excluson of spatal structures and nonperpendcular loads s justfed n that the methods shown here must have the basc prerequste of decomposablty of the entre system nto a statcally determnant component system. Dstnctons shall be made between the followng herarchcal systems: Sngle bar (a n Fg. ): smallest system unt. Basc system (b n Fg. ): decomposable nto sngle bars. Component system (c n Fg. ): decomposable nto basc systems and sngle bars. Entre system (Fg. ): decomposable nto component systems, basc systems and sngle bars. The ndvdual systems can be combned wth each other. Ths results n herarchcally equal or hgher system types. The connecton wthn the system s defned by node ponts. In ths vew, the load of a system s seen not as a statc equlbrum state n whch actng forces are establshed from the supportng forces, but the load s consdered to be an teratve process of the nteracton of sub-systems. In ths connecton, each teraton step s representatve of the statc equlbrum state observaton of sub-systems. The Structural Engneerng Internatonal /2 Scentfc Paper 8
5 7 Fg. : Entre system wth sub-systems. top left: Sngle bar; bottom left: Basc system; (c) rght: component system ntal state of the teraton process s the external load of the system. In each teraton step, the supportng forces from the condtons of equlbrum result from the progressons of the observed sub-system. These n turn represent n the next teraton step the progressons of the observed neghbourng system. s the process progresses, the number of observed sub-systems ncreases (a n Fg. 4). Begnnng at any startng pont or at any teraton step, the analyss of the teraton step progressons leads to a dfferentaton of two dfferent types of teraton step progressons (b n Fg. 4). Cyclcal: Progresson where any bar s nvolved n the observaton recurrently each tme after a specfc number of teraton steps. bar n a cyclcal progresson thus demonstrates a concatenaton of nteractons wth tself. The entrety of bars observed wthn one cyclcal progresson s defned as one possble basc system. Dffused: Progresson where each bar s nvolved n the observaton only once, 2 (c) 8 2 9 thus no bar exhbts a concatenaton of nteractons wth tself. Such progressons descrbe the dsperson of the teraton steps n the system and descrbe the lnkng of the basc system wth ts neghbourng system. Behavour of a Basc System Frst of all, the teratve process wll be analysed on a basc system, based on a decomposton nto sub-systems. Here, the sub-systems are sngle bars, representng the smallest system unt of a recprocal frame structure. In the followng, n s the number of ndvdual bars consttutng the basc system whereby n must be >, as otherwse no operatng system s possble. Furthermore, k s the ndex of the observed bar and that of the present teraton step. K,k and K B,k are the node ponts of the system, whereby K,k forms the support ponts on the system edge and K B,k the contact ponts to the neghbourng bar. k and B k are the respectve nodal forces. a k and b k apply to the proportons of node dstances on the sngle bar, k. Fg. 4: left: Example of three teraton steps: frst red, second blue, thrd green; rght: Example of a cyclcal and a dffused teraton step progresson 4 6 a b k k KBk, KBk, = K K k, Bk, Kk, KBk, = K K a k + b k = k, Bk, () The condton of equlbrum results n the sub-system of the sngle bar k from the actng force B k : k = a k. B k (2) B k = b k. B k s t s a matter of cyclcal observaton, the ndex k = s the equvalent of k = n, and k = n s the equvalent of k =. Therefore, the varables u = (k + )modn and v = (k + )modn are ntroduced. Thus, f B k s the actng force on the basc system, u and B u apply to the ncrease n statc forces at ponts K,u and K B,u n any teraton step >. u = a u. B v () B u = b u. B v Each bar of the basc system s nvolved exactly once n an equlbrum formulaton wthn a progresson of n teraton steps, whereby the node forces k and B k receve a value each tme, whch s the ncrease n statc force at nodes K,k and K B,k n each teraton step. Fgure 6 llustrates the values B k n a regular four-barred basc system wth one actng force at node pont K B,. It shows an exponentally decreasng ncrease n statc forces at the nodes. The equlbrum formulaton of all node ponts can be descrbed n one term through matrx notaton. Thus, one step of the teratve process for any basc system made up of n bars can generally be formulated as follows: F = F (4) where, denoted n the followng as the load dstrbuton matrx, ncludes the geometrcal condtons of all bars n a planar basc system (e.g. see Fg. 5). a b a b = a2 b2 a n b n R 2n 2n (5) 82 Scentfc Paper Structural Engneerng Internatonal /2
K, K, K B,2 K B, B 2 K B, K,2 K B, B B K, F relates to the begnnng of the teratve observaton, and comprses the external forces of the system. It apples to the ncrease of node forces n teraton step : F = F (7) Wth the maxmum norm < apples lm F = (8) F t apples to the node forces exstng n teraton step t. These are the sum of all force ncreases of the precedng teraton steps: S, 667 S S F =, 667 S S, 667 S () The components k = S (support ponts of the system) and B k =,667. S (contact ponts to the neghbourng bars) ncluded n F are equvalent to the node forces resultng from a statc observaton of ths system n the classcal sense. Fg. 5: Example of a basc system (n = 4). statcs system; sub-system bar [%] 9 8 7 6 5 4 2 2 4 5 6 7 8 9 2 4 5 ( ) Fg. 6: Iteraton process of B k wth n = 4, a k =,4, b k =,6 F (n Eq. 4) ncludes all the ncreases n node forces n the observed teraton step : F, B,, B, = n, B n, F R 2n (6) B B B B B 2 F = t t = F (9) F= lm t Ft results wth the maxmum norm < : F= F = F = E F ( ) = = dentty matrx E R 2n 2n () The node forces F determned n ths manner (Eq. ) correspond to the values of a statc equlbrum observaton n the classcal sense. s an example, a three-barred basc system as seen n Fg. 2c s consdered. The proportons of the node dstances n all bars are a =, and b =,7. Ths results n a load dstrbuton matrx as follows:,,7, =,7,, 7 () For the external load, only dead load s consdered. Therefore the load n each node pont s half of the bar weght, g =,5. S. Wth t apples, 5 S, 5 S, 5 S F =, 5 S, 5 S, 5 S (2) The lmt of the teratve observaton as shown above (Eq.) results n the followng: Systemc structure of recprocal frame structures To comprehend the teraton process of an entre recprocal frame structure as descrbed above, t s necessary to formulate the load dstrbuton matrx of the entre system. Usng a nodewse process, as shown n the prevous secton, can become very complex. For ths reason, the followng shows the analoges between herarchcal structures of systems accordng to the defnton gven at the begnnng and accordng to the structure of the load dstrbuton matrx from sub-matrces. Ths nformaton allows for the systematc creaton of the load dstrbuton matrx for a complex recprocal frame system. The structure takes place accordng to the herarchy defned above. Sngle Bar When observng an ndvdual bar of a recprocal frame system, four node ponts can be specfed where forces may occur: Two ponts K F, and K F,, where the bar les on ts neghbourng bars or wth the support of the entre structure. Two ponts K F, and K F,2, whch n turn form the support for other bars and, as a result, actng forces can result at these ponts. a, b, c and d apply to the proportons of node dstances of a sngle bar (Fg. 7). KF,2 KF, KF, KF,2 a = b = (4) K K K K a+ b= F, F, F, F, Structural Engneerng Internatonal /2 Scentfc Paper 8
F F 2 K F, K F, + + + + K F, K F,2 F Fg. 7: Node ponts and forces of a sngle bar KF, KF, KF, KF, c = d = (4) K K K K F, F, F, F, F 2 2 Bar Bar Bar Bar Fg. 8: Example n = 4: System wth ndependent bars; system wth dependent bars c+ d= nalogous to what was shown n the prevous secton, an teraton step can be descrbed as follows: F = F S, S,S wth S F, S c a = d b F F = F F,, 2,, (5) Snce the system s a statcally determnate system, the elements of the load dstrbuton matrx s consst only of geometrcal factors. F,s comprses the node forces at the begnnng of the teratve observaton. F,s apples to the sngle bar for >. FS, = Basc System The basc system s bult from n sngle bars. The followng ntends to demonstrate that the system s load dstrbuton matrx G can be set up from a structure of n n sub-matrces. For that purpose, we must note n a frst step the n load dstrbuton matrces s,k of the sngle bars on the man dagonals of the basc system s matrx. G S, S, = Sn, (6) s only the man dagonals are beng used, there s no nteracton of the submatrces n ths matrx structure. Ths corresponds to a system made up of n ndependent bars (Fg. 8a). To acheve a system wth dependent bars (Fg. 8b), sub-matrces must be produced n the load dstrbuton matrx on the other sde of the man dagonals that descrbe the dependence of the sngle bars. These submatrces are descrbed n the followng text. The connectng of two bars corre - sponds to the overlayng of two node ponts. Ths results n a mergng together of the two columns assgned to the nodes lsted n the load dstrbuton matrx G. The assgnment of node g on bar k n the system to column h of matrx G s h=4 ( k ) + g (7) Should any node g from bar k be overlad wth node g from bar k, ths corresponds to the transposton of correspondng columns h and h of G (Fg. 9a), as well as to the subsequent removal of column h and of lne h (Fg. 9b). s an example, ths overlayng of two nodes s shown n a basc system made up of four bars (Fg. 9). If ths method s used to proceed wth all four connecton nodes, the basc system results (Fg. 8b), whch correspond to the followng load dstrbuton matrx: G c a d b a c b d = a c 2 2 b2 d2 a c b d (8) G (Eq. 8) s functonally dentcal to the load dstrbuton matrx of the basc system shown n the prevous secton (Eq. 5). However, n G the non-loaded node ponts are taken nto consderaton and n the followng become sgnfcant when the basc system s expanded. Component System In the same way that a basc system s constructed from sngle bars, the formaton of a sutable component system s made up of dentcal or nondentcal basc systems. n observaton G G c a d b c a h h c a d b a c d b h b d c 2 a 2 d 2 b 2 c a d b Fg. 9: Transposton of columns h and h ; removal of column h and lne h h c 2 a 2 d 2 b 2 c a d b 84 Scentfc Paper Structural Engneerng Internatonal /2
of component systems s useful f patterns repeat n a recprocal frame structure. To comprehend a component system made up of m basc systems, the load dstrbuton matrces of the basc systems have to be placed on the man dagonals of the component system s matrx. Ths corresponds to m ndependent basc systems (Fg. a). T G, G, = Gm, (9) To establsh a connecton of the basc system (Fg. b), sub-matrces must be produced on the other sde of the man dagonals of T whch descrbe the nteracton of the dagonal elements. Ths takes place analogous to the process that was descrbed for the constructon of the basc system. s an example, the followng s the load dstrbuton matrx results for a component system made up of four basc systems: T G, B C C G B =, C2 G,2 B2 B C G, (2) For ths component system, two dfferent types of sub-matrces result. B descrbes the connecton to the basc element joned n a counter-clockwse manner; C s the connecton to the basc element whch jons n a clockwse manner. Fg. : Component system wth ndependent basc systems; component system wth dependent basc systems For systemc observaton, dstnctons were made at the begnnng of the text between cyclcal and dffused progressons of teratve steps. The load dstrbuton matrx can be lkewse decomposed nto a cyclcal part T,z and a dffused part T,d. It s vald Tz, G, G, = = Td, T Tz, Gm, (2) These parts, for the example shown above, are Tz, Td, G, G,2 = G, B C C B 2 2 = C B B C 4 4 G,4 (22) In each of the basc systems from whch the component system s made, one cyclcal progresson of teraton steps can be determned. Ths progresson s descrbed by the cyclcal part T,z. Each sub-matrx G,p ( p < m) here descrbes those progressons of teraton steps that comprse exclusvely components of the accompanyng subsystem p. T,d descrbes the dffuson of the teratve observaton, and thus those progressons of teratve steps that descrbe the nteracton of the ndvdual sub-systems. The man matrx T and ts dagonal elements G,p must be quadratc; however, the sub-matrces of the dffused part must not. Entre System In the same manner that component systems are constructed from basc systems, entre systems can be bult from basc and component systems. For ths, the results of the load dstrbuton matrx, whch regardng constructon and characterstcs, s comparable to the load dstrbuton matrx of a component system. s also decomposable nto a cyclcal part z and a dffused part d, both of whch behave smlar to parts T,z and T,d for a component system. In order to verfy the method of teratve observaton of element nteracton, the results of dfferent structures were compared wth those of a statc calculaton through the fnte element software Cubus. s an example, the comparson for the entre system shown n Fg. s llustrated here. It s loaded wth a force of kn at node 2 and has proportons of node dstances of a =, and b =,7 n all bars. Table shows the results of the supportng ponts (nodes to ) n both procedures. It can be seen that the results dffer from each other by less than %. Concluson In ths paper, a possblty has been descrbed to examne the forces wthn plane recprocal frame structures. The llustrated approach s not meant to be a practcal desgn procedure but s to be consdered as merely academc. In the frst place, t s to develop an understandng of the behavour of such structures and t can therefore form the bass of practcal methods that analyse and desgn recprocal frames or comparable structures. The method s a systematc analyss of the nteractve behavour of the totalty of all sub-systems n a plane recprocal frame structure. nalogous to constructon from sub-systems, constructon of the load dstrbuton matrx, whch s necessary for the observaton shown here, s a result of sub-matrces. s such, sub-systems and recprocal frame structures can be of any form f constructon of the bars s geometrcally compatble and the structure can functon accordng to the recprocal frame prncple. By usng the teratve observaton method to observe the ncrease n node Node no. 2 4 5 6 7 8 9 Iteratve (kn) 6,6,2 8,4 2,4,44 8,5,82,48 2,,4 2,76,49 Fnte Element 6,57,4 8,9 2,2,44 8,22,82,5 2,4,2 2,78,49 Method (kn) Dfference (%),9,6,2,98,9,96,4,5,72 Table : Evaluaton of the proposed method by the Fnte Element Method Structural Engneerng Internatonal /2 Scentfc Paper 85
forces wthn the system, t s possble to llustrate the systemc behavour of the nteracton of sub-systems n recprocal frame structures as follows: F = F (7) F F 2 F F By llustratng the ndvdual teraton steps, t s possble to produce a smulaton comparable to the load dstrbuton n a structure. Moreover, the lmt of the sum of all teraton steps can be unformly formulated as follows: ( ) () F = E F Fg. : Elements of a grd K, K, F Fg. : Geometry and force dagram of a sngle bar n general case F F 2 F K F K, K,2 Fg. 2: Number of teraton steps untl the support pont s ncluded: d(k F,K, ) = 4, d(k F,K, ) = 7, d(k F,K,2 ) =, d(k F,K, ) = 7 Ths lmt F conssts of all node forces of the structure accordng to the statc equlbrum. comparable systemc observaton can also be appled to varous other structures. However, the prerequste exsts that the structures must be made up of dscrete elements and that the sub-system of each element must be assumed to be a statcally determnate equlbrum system. For nstance, ths method can be used to determne the lateral forces at the node ponts of grllages, assumng that all loads act perpendcular to the grllage. decomposablty nto elements must be found that nteract on the prncple of recprocal frames, so that each sub-system (Fg. ) can then be assumed to be a statcally determnate equlbrum system. In an easy manner, the nternal forces of a grd can be found through ths method and, out of ths, smple geometrcal rules for the formaton of the ndvdual bars can be establshed so that they meet the statc demands. In the llustrated example (Fg. 2), the ndvdual steps of an teratve observaton of sub-systems on a grllage are shown startng at the actng force F at the node pont K F. The fgure shows the smallest possble number of teraton steps for each of the four support ponts K,w ( w < 4) up to the pont that t s nvolved n the observaton for the frst tme. Snce the forces observed n each step decrease exponentally as the teraton progresses (Fg. 6), the method demonstrates that the forces n a load-bearng structure are prmarly carred by the support whch exhbts the smallest dstance to the actng force. If the structural condtons of a slab that s beng acted upon by perpendcular forces are llustrated wth approprate grllage, ths method can also be appled. Further developmental potental of the method shown conssts of, n the smplest case, the expanson of spatal structures under non-perpendcular-actng loads. s such, comparable behavour n the nteracton of subsystems s to be expected, wth the occurrence of systemc behavour smlar to that shown here. The challenge here s the statc ndetermnacy of the sub-systems (Fg. ). The support forces F and F can no longer be determned solely wth geometry and wth the actons F and F 2, but addtonal assumptons, for example, of materal or jonts are necessary. Yet another challenge s the transmsson of the systemc observaton shown here onto smlarly behavng spatal structures made up of dscrete elements. These structures are of partcular sgnfcance n the area of bomorphc archtecture. Contemporary examples are the Olympc stadum n Bejng by Herzog and de Meuron and the Centre Pompdou n Metz by Shgeru Ban. References [] D Carlo B. The wooden roofs of Leonardo and new structural research. In Nexus Network Journal, Km Wllams Books: Turn, 28; 27 8. [2] Popovc Larsen O. Recprocal Frame rchtecture. rchtectural Press, Elsever: Oxford, 28. [] McQuad M. Shgeru Ban, Phadon Press Lmted: New York, 2. [4] de Honnecourt V. lbum de Vllard de Honnecourt. Laget: Pars, 976. [5] Pedrett C. LEONRDO rchtetto. Electa Edtrce: Mlano, 978. [6] Hart V, Hcks P. Sebastano Serlo on rchtecture. Yale Unversty Press: New Haven, 996. [7] Wnter K, Rug W. Innovatonen m Holzbau De Zollnger Bauwese, Bautechnk 4/992. Ernst & Sohn: Berln, 992; 9 97. [8] Hensel M, Menges. Morpho-Ecologes, rchtectural ssocaton: London, 26. [9] Chlton JC, Choo BS, Yu J. Morphology of recprocal frame three-dmensonal grllage structures. In Spatal Lattce and Tenson Structures of the ISS-SCE Internatonal Symposum. mercan Socety of Cvl Engneers: New York, 994; 65 74. [] Bertn V. Hebelstabwerke, RCH+, vol. 59/6. RCH+ Verlag: achen, 22; 5 5. [] Baverel O, Sadan M. The mult-recprocal grd system. J. Int. ssoc. Shell Spatal Struct. 999; 4: 4. [2] Sánchez J, Escrg F. daptable Leonardo. Internatonal Conference On daptable Buldng Structures, Endhoven, 26; 28 2. [] Baverel O, Douthe C, Caron J-F. Nexorade: a structure for free form archtecture. Internatonal Conference on daptable Buldng Structures Endhoven, 26; 76 8. 86 Scentfc Paper Structural Engneerng Internatonal /2